On the definition of geometric Dirac operators

Abstract

In this largely expository paper we give a self-contained treatment of the Dirac operator. Emphasizing the algebraic point of view we first sketch the necessary prerequisites from Clifford algebras and their representations and then define (and characterize) spin structures and the corresponding Spin-Dirac operator purely in the spirit of irreducible representations. We also prove its equivalence with the usual approach via principal bundles. We conclude with other geometric Dirac operators and their properties using the afore-mentioned setting.